264 J. Opt. Soc. Am. A / Vol. 30, No. 2 / February 2013 Masaoka et al.
Number of discernible object colors is a conundrum
Kenichiro Masaoka,1,2,* Roy S. Berns,2 Mark D. Fairchild,2 and Farhad Moghareh Abed2 1NHK Science & Technology Research Laboratories, Tokyo 157-8510, Japan 2Munsell Color Science Laboratory, Chester F. Carlson Center for Imaging Science, Rochester Institute of Technology, Rochester, New York 14623, USA *Corresponding author: masaoka.k‑[email protected]
Received June 15, 2012; revised December 19, 2012; accepted December 24, 2012; posted January 2, 2013 (Doc. ID 169396); published January 31, 2013 Widely varying estimates of the number of discernible object colors have been made by using various methods over the past 100 years. To clarify the source of the discrepancies in the previous, inconsistent estimates, the number of discernible object colors is estimated over a wide range of color temperatures and illuminance levels using several chromatic adaptation models, color spaces, and color difference limens. Efficient and accurate models are used to compute optimal-color solids and count the number of discernible colors. A comprehensive simulation reveals limitations in the ability of current color appearance models to estimate the number of discernible colors even if the color solid is smaller than the optimal-color solid. The estimates depend on the color appearance model, color space, and color difference limen used. The fundamental problem lies in the von Kries-type chromatic adaptation transforms, which have an unknown effect on the ranking of the number of discernible colors at different color temperatures. © 2013 Optical Society of America OCIS codes: 330.1730, 330.4060, 330.5020, 120.5240, 230.6080, 300.6170.
1. INTRODUCTION he modified his estimate to 32,820 visual sensations (660 + The number of discernible object colors is informative not 160 + 32,000) in his revised book [4]. In 1939, Boring et al. only because of scientific interest in human color vision [5] explained that there are 156 just-noticeable differences but also because of its potential use in industrial applications (JNDs) in hue, 16–23 JNDs in saturation, and 572 JNDs in such as the development of pigments and dyes, evaluation of brightness, but he somehow made an inflated estimate of the gamut coverage of displays, and design of the spectral the number of discriminably different colors, on the order power distribution of light sources. The color gamut of color of 300,000, which is comparable to his estimate of the number imaging media and its main controlling factor, the viewing of discernible tones that differ either in pitch or in loudness. conditions, are of significant practical as well as theoretical He said, “In a sense there are, speaking generally, about as importance in the reproduction of color images [1]. However, many tones as colors.” Also in 1939, Judd and Kelly [6] esti- widely varying estimates of the number of discernible colors mated that about 10 million colors are distinguishable in day- have been made over the past 100 years. This motivated us to light by the trained human eye, although they did not mention clarify the source of the variation and derive a more accurate the grounds for their estimate and other details (e.g., criteria estimate. for color differences, assumed illuminant). This figure of 10 Perceptible color is represented in a color solid on the million was also used in the book by Judd and Wyszecki basis of the trichromatic nature of human vision, where a [7]. In 1951, Halsey and Chapanis [8] stated that a normal color solid is a three-dimensional representation of a color observer can discriminate well more than 1 million different model. Perceptual color space can be reasonably represented colors under ideal, laboratory conditions of observation. in cylindrical coordinates (hue, chroma/colorfulness, and However, in 1981, Hård and Sivik [9] stated that our ability lightness/brightness) or Cartesian coordinates (two opponent- to identify a given color with some certainty is a great deal color dimensions and lightness/brightness). The number of less and would probably cover only about 10,000–20,000 discernible colors contained in a three-dimensional solid in colors. a perceptual color space has been estimated using different In addition to these results, recent estimations were made methods. Kuehni [2] has summarized the early research. using optimal colors (i.e., optimal-color solids). An optimal Unfortunately, some of these studies do not provide sufficient color is the nonfluorescent object color having the maximum detail. To make matters worse, some estimates by authorities saturation under a given light. Ostwald [10] empirically found have been used repeatedly in other journal papers and books that optimal-color reflectances are either 0 or 1 at all wave- without reference to the original estimates, which makes it lengths with at most two transitions. The first mathematical difficult to identify the authorities. Here, we investigate the proof of this, which uses the convexity of the spectrum locus, original estimates in detail, referring to Kuehni’s summary, is attributed to Schrödinger [11]. Following the calculation of and add some recent estimates, as summarized in Table 1. optimal colors by Luther [12], Nyberg [13], and Rösch [14], In 1896, Titchener [3] experimentally identified 30,850 MacAdam generated a geometric proof of the optimal-color visual sensations (700 “brightness qualities” + 150 “spectral theorem using the CIE xy chromaticity diagram [15] and cal- colors” + 30,000 “color sensations of mixed origin”). Later, culated the chromaticity coordinates of the optimal-color loci,
1084-7529/13/020264-14$15.00/0 © 2013 Optical Society of America Masaoka et al. Vol. 30, No. 2 / February 2013 / J. Opt. Soc. Am. A 265
Table 1. History of the Estimates of the Number of Discernible Colors
Year Researcher Estimate Illuminant Color Data/Color Model 1896 Titchener [3] 30,850 700 “brightness qualities” + 150 “spectral colors” + 30,000 “colors mixed with origin” 1899 Titchener [4] 32,820 660 “brightness qualities” + 160 “spectral colors” + 32,000 “colors mixed with origin” 1939 Boring et al. [5] 300,000 156 hue × 16–23 saturation × 572 brightness 1939 Judd and Kelly [6] 10,000,000 Unknown 1943 Nickerson and Newhall [17] 7,295,000 Optimal color solid/Munsell notation 1951 Halsey and Chapanis [8] 1,000,000 Unknown 1981 Hård and Sivik [9] 10,000–20,000 Unknown 1998 Pointer and Attridge [18] 2,280,000 D65 Optimal color solid/CIELAB 2007 Wen [19] 352,263 D65 Optimal color solid/CIE94 2007 Martínez-Verdú et al. [20] 2,050,000 E Optimal color solid/CIECAM02 lightness–colorfulness (JaM bM ) 2,046,000 C 2,013,000 D65 1,968,000 F7 1,753,000 A 1,735,000 F11 1,665,000 F2 2008 Morovic [22] 1,900,000 D50 Optimal color solid/CIECAM02 lightness–chroma (JaC bC ) 2012 Morovic et al. [23] 4,200,000 F11 Optimal color solid/CIECAM02 lightness–chroma (JaC bC ) 3,800,000 D50 3,500,000 A 1,700,000 D50 LUTCHI/CIECAM02 lightness–chroma (JaC bC)
called the MacAdam limits, under illuminants A and C, with and color space, in 2012, Morovic et al. [23] doubled the num- luminous reflectance Y for the third dimension [16]. ber to 3.8 million colors under illuminant D50, in addition to In 1943, Nickerson and Newhall [17] applied the Munsell other estimates of 4.2 million under illuminant F11 and 3.5 notation to the MacAdam limits and identified a total of million under illuminant A. On the other hand, Morovic et al. 5836 full chroma steps for 40 hues spaced 2.5 hue steps apart, also pointed out that the currently available models failed to and nine values spaced one value step apart. They estimated give predictions when extrapolating the psychophysical data the number of discriminable colors as 7,295,000, assuming they are based on; finally, they made an alternative, safe es- that one chroma, hue, or value step corresponds to 5, 2, and timate of at least 1.7 million colors, which was their estimate 50 JNDs, respectively. In 1998, Pointer and Attridge [18] of the color gamut volume of the LUTCHI [24] data used to counted the unit cubes within an optimal-color solid under build CIECAM02. illuminant D65 in the CIE 1976 L ;a ;b color space Unfortunately, it is unknown which estimate is the most (CIELAB) and made an estimate of about 2.28 million. In reliable. The validity of even the estimates based on the color 2007, Wen [19] sliced an optimal-color solid under illuminant appearance models [18–20,22,23] has not been verified. To D65 at each lightness unit and divided each slice into pieces complicate matters, the recent estimates were made using having the unit chroma and hue differences described in different combinations of color appearance models (CIELAB CIE94, obtaining a count of 352,263. Also in 2007, Martínez- and CIECAM02), color spaces (lightness–chroma and Verdú et al. [20] counted unit cubes packed in optimal-color lightness–colorfulness), color difference limens (CIELAB unit, solids under several standard illuminants (A, C, D65, E, F2, F7, CIE94 color difference, and CIECAM02 unit), and lighting con- F11) and high-pressure sodium lamps in the lightness and ditions (e.g., illuminants D50 and D65). It is important to clarify colorfulness space J;aM ;bM of the CIECAM02 color appear- the source of the discrepancies in previous, inconsistent ance model [21]. They assumed this to be the most uniform estimates and determine the limitations of the current color ap- color space on the basis of their observation that the calcu- pearance models in both a theoretical and practical sense. lated optimal-color solids in the CIECAM02 color space were In this paper, we provide estimates of the number of dis- relatively spherical compared to those in the CIELAB color cernible colors within the optimal-color solid for a wide range space and others. They estimated that there are 2.050 million of color temperatures and illuminance levels, using a fast, ac- distinguishable colors under illuminant E, 2.046 million under curate model for computing the optimal colors. Several chro- illuminant C, 2.013 million under illuminant D65, 1.968 million matic adaptation models, color spaces, and color difference under illuminant F7, 1.753 million under illuminant A, and few- limens are used to verify the recent estimates based on color er colors under the other light sources. They also suggested appearance models. the possibility of an alternative color-rendering index based on the number of discernible colors within an optimal-color 2. METHODS solid. In 2008, Morovic [22] calculated the convex hull volume Figure 1 shows a flowchart of the method used to compute an of optimal-color solids in the lightness and chroma space optimal color with a specified central wavelength under an J;aC;bC of CIECAM02 and estimated a total of 1.9 million illuminant having a given spectral power distribution and colors under illuminant D50. Using the same counting method adaptation illuminance level. Optimal colors were searched 266 J. Opt. Soc. Am. A / Vol. 30, No. 2 / February 2013 Masaoka et al.
Fig. 1. Flowchart for computing an optimal color. iteratively so that their lightness values were equalized to the The spectra of the daylight illuminants were calculated using given lightness values. To estimate the number of discernible the CIE equation [29]. Following the recommendation of CIE colors within an optimal-color solid, we computed the [30], spline interpolation was used for the CIE color-matching optimal-color loci at regular lightness unit intervals of L in functions and daylight illuminant spectra in order to ensure a CIELAB, J in CIECAM02, and J0 in a CIECAM02-based uni- sufficiently small wavelength step of 0.1 nm. The blackbody form color space (CAM02-UCS) [25]. Three different chro- radiation was calculated using Planck’s law at wavelength in- matic adaptation transformations (CATs) were embedded tervals of 0.1 nm. The light source spectrum S was normalized ΣN S k y¯ k 100 N in front of CIELAB with a D65 reference white point. Black- so that k 1 , where is the number of wave- body radiators at color temperatures ranging from 2000 to length steps, N 3001. 4000 K and standard daylight illuminants at CCTs ranging To avoid switching between types I and II during optimiza- from 4000 to 10,000 K at 500 K intervals were used as light tion of λcut-on and λcut-off , three copies of the color-matching sources in the simulation. Illuminance levels of 200 lx functions x¯ k , y¯ k , and z¯ k , and illuminant spectrum S k (museum standard [26]), 1000 lx (reference viewing condition were each concatenated separately, forming four sequences [27]), 10,000 lx (outside on cloudy days), and 100,000 lx denoted respectively as x¯ c, y¯ c, and z¯c, and Sc, as shown in (outside on sunny days) were considered for the reference the upper part of Fig. 3, where n is the central wavelength, white in CIECAM02. Additionally, illuminant E and illuminant and hn is the half bandwidth of the spectral reflectance series F (F1–F12) were used at an illuminance of 1571 lx to Rn l of the optimal color on the concatenated wavelength l l n − h l n h verify the results of Martínez-Verdú et al. [20]. scale . Here, cut-on n, and cut-off n: 1;l ≤ l ≤ l A. Optimal-Color Computation cut-on cut-off Rn l ; (1) Optimal colors are generated using stimuli with either 0 reflec- 0; otherwise tance (transmittance) at both ends of the visible spectrum and 1 in the middle (type I) or 1 at the ends and 0 in the middle where n is an integer between N 1 and 2N, and hn is a real number between 0 and N∕2. It is obvious that concatenation (type II). Figure 2 illustrates both types, where λcut-on and makes it possible to treat a type II band-stop reflectance as a λcut-off represent the cut-on and cut-off wavelengths, respec- tively. In our simulation, the end colors (high-pass and low- pass) are handled as type I, with the wavelength of transition (λcut-on or λcut-off ) being the end point of the visible spectrum. A modified version of Masaoka’s model [28] was used to calculate the tristimulus values of optimal colors. We used the CIE 1931 color-matching functions x¯, y¯, and z¯ at wavelengths ranging from 400 to 700 nm at 1 nm intervals.
Fig. 3. (Color online) Concatenation of three copies of color- matching functions and illuminant spectrum (top). An optimal color Fig. 2. (Color online) Two types of spectral reflectance (transmit- has spectral reflectance R l with central wavelength n and half band- tance) for optimal colors. width hn on the concatenated wavelength scale l (center and bottom). Masaoka et al. Vol. 30, No. 2 / February 2013 / J. Opt. Soc. Am. A 267 type I band-pass reflectance. Let T x l , Ty l , and T z l be value between 0 and 1 can be set, which enables the resulting continuous functions obtained by linear interpolation of loci to be smoother than those obtained by Martínez-Verdú Scx¯ c, Scy¯ c, and Scz¯ c, respectively. Figure 4 shows a diagram et al.’s method, despite the sparse wavelength intervals. illustrating trapezoidal integration of T l . The tristimulus va- The spectral reflectance including a non-zero and non-one lues of the optimal color can be efficiently calculated using value is, however, no longer that of the optimal color by trapezoidal integration: definition. The tolerances of the optimized parameters are un- Z Z known. The 1 nm wavelength intervals degrade the accuracy l cut-off of optimal-color computation under spiky-spectrum illumi- Xn R l Tx l dl Tx l dl nants. On the other hand, our method optimizes λ and lcut-on cut-on λcut-off for an optimal color on a continuous scale rather than T x n − ⌊hn⌋ − 1 · fhng T x n − ⌊hn⌋ finding discrete reflectance values. Prevention of switching · 2 − fhng · fhng∕2 between type I and type II dramatically reduced the computa-
T x n ⌊hn⌋ 1 · fhng Tx n ⌊hn⌋ tional cost; it took only about 10 s to obtain each locus. This computational efficiency enabled us to compare the estimates 2 − h h ∕2 · f ng · f ng over a wide range of color temperatures and illuminance n X⌊hn⌋ levels using several color models. T x k − Tx n − ⌊hn⌋ ∕2 − Tx n ⌊hn⌋ ∕2: (2) k n−⌊hn⌋ B. Color Appearance Model We used CIELAB with three different CATs to verify the es- ⌊h ⌋ h h Here n is the floor of n, and f ng is the fractional part of timates of Pointer and Attridge [18]. In CIELAB, the vertical h n. The first two terms in Eq. (2) represent the end margins of dimension represents the lightness L , which ranges from 0 to the integral, and the other terms represent trapezoidal integra- 100; the two horizontal dimensions represent the opponent n − ⌊h ⌋ n ⌊h ⌋ Y Z tion from n to n . n and n are described by red/green channel a and yellow/blue channel b . The CIELAB T T T replacing x in Eq. (2) with y and z, respectively. The es- color space can also be represented in cylindrical coordinates timated tristimulus values of an optimal color were converted using the chroma C and hue hab. A model that contains pre- L ab to the lightness value n by using a specified color appearance dictors of at least the relative color appearance attributes of h L model. Here, n was optimized by a simplex method so that n lightness, chroma, and hue is referred to as a color appear- L becomes equal to a specified lightness value const with an ac- ance model [32]. In that sense, CIELAB can be considered 10−7 h curacy of . The termination tolerance for n was set to a color appearance model, although the adaptation transform, 10−10 10−11 on the concatenated wavelength scale or nm. which normalizes the stimulus tristimulus values by those of The MATLAB code we used is presented in the appendix. the reference white, is clearly less accurate than transforma- The MATLAB fminsearch function was used to search for tions that follow known visual physiology more closely [33]. h L − L a local minimizer n of the function j const nj in a specified The CIE does not recommend that CIELAB be used when the N∕2 range from 0 to . illuminant is “too different from that of average daylight” [34]. This method is more computationally efficient and accurate To investigate how that inaccuracy affects the estimate of the than other methods. In the method of Martínez-Verdú et al. number of discernible colors, we embedded some chromatic [20], the algorithm computes optimal colors with all the pos- adaptation models based on the von Kries hypothesis [35]in λ λ sible pairs of cut-on and cut-off at 0.1 nm wavelength intervals the CIELAB equations. and looks for the optimal colors whose lightness values are The von Kries-type CATs linearly convert tristimulus values equal to a given lightness with a given tolerance. They to relative cone responses and scale them so that those of the suggested 0.01 for the lightness tolerance in order to obtain reference white stay constant for both the destination and the adequate samples of optimal colors to form a locus. However, source conditions. The long-, medium-, and short-wavelength the lightness tolerance obtained using such a trial-and-error cone responses, L, M, and S, are transformed from the X, Y, method is not always appropriate for an arbitrary illuminant. and Z values by using a 3 × 3 matrix M. The matrix notation Another problem is the computational cost. It took about 1 h can be extended to the calculation of the corresponding for each optimal-color type and lightness. In the method of colors across two viewing conditions and to explicitly include Li et al. [31], the algorithm finds reflectance values at intervals the transformation from tristimulus values to relative cone of 1 nm instead of 0.1 nm. The computational cost is less than responses as follows: that of Martínez-Verdú et al.’s method; it took several minutes to obtain a locus. The discrete reflectance values are either 1 2 3 2 3 2 3 Xd Lw;d∕Lw;s 00Xs or 0 at all wavelengths except for λcut-on or λcut-off , where a 4 5 −14 5 4 5 Y d M 0 Mw;d∕Mw;s 0 M Y s ; Zd 00Sw;d∕Sw;s Zs (3)
where subscripts d and s denote the destination and the source, respectively, and Lw, Mw, and Sw are the long-, med- ium-, and short-wavelength cone responses, respectively, of the reference white. The normalization in CIELAB can be expressed by letting M be the 3 × 3 identity matrix. We used three different von Kries-type CATs: CAT02, Fig. 4. (Color online) Diagram of trapezoidal integration of T l . Hunt–Pointer–Estévez (HPE), and the linearized version of 268 J. Opt. Soc. Am. A / Vol. 30, No. 2 / February 2013 Masaoka et al. the Bradford CAT (BFD). CAT02 converts CIE tristimulus were converted to the HPE responses L0, M0, and S0. The 0 0 0 values to RGB responses with the “sharpened” cone respon- post-adaptation cone responses La, Ma, and Sa were modified sivities, which are spectrally distinct and partially negative, to avoid calculating the power of negative numbers: whereas the HPE fundamentals more closely represent actual 0 0 0 0.42 cone responsivities [33]. CAT02 is used for chromatic adapta- L ∕jL j · 400 F LjL j∕100 L0 0 1; a 0 0 0 0.42 . (12) tion in CIECAM02, whereas HPE is used for post-adaptation. 27.13 L ∕jL j · 400 F LjL j∕100 BFD is commonly used for the profile connection space (PCS) required for International Color Consortium profiles in color where F L is the luminance-level adaptation factor: management [36]. The 3 × 3 matrices for CAT02 (MCAT02), 4 4 2 1∕3 HPE (MHPE), and BFD (MBFD) are described as follows: F L 0.2k 5LA 0.1 1 − k 5LA ; (13) 2 3 0.7328 0.4296 −0.1624 4 5 k 1∕ 5LA 1 : (14) MCAT02 −0.7036 1.6975 0.0061 ; (4) 0 0030 0 0136 0 9834 . . . 0 0 0 0 Ma and Sa are described by replacing L in Eq. (12) with M and S0, respectively. These modifications were necessary be- 2 3 cause the bandwidth of the spectral reflectance R l of opti- 0 38971 0 68898 −0 07868 . . . mal colors can become extremely narrow during optimization, M 4 −0 22981 1 18340 0 04641 5; HPE . . . (5) producing negative responses. The lightness J, chroma C, 0.00000 0.00000 1.00000 colorfulness M, and hue h were then computed. The ratio of colorfulness M to chroma C was calculated as 2 3 0.8951 0.2664 −0.1614 M C F 1∕4: 4 5 · L (15) MBFD −0.7502 1.7135 0.0367 : (6) 0.0389 −0.0685 1.0296 The Cartesian coordinates for the chroma aC;bC and color- fulness aM ;bM are C cos h ;C sin h and M cos h ; We used CIECAM02 and CAM02-UCS to verify the esti- M sin h , respectively. mates of Martínez-Verdú et al. [20], Morovic [22], and Morovic CIECAM02 does not necessarily assume a color space that et al. [23]. CIECAM02 is the latest color appearance model re- is perceptually uniform in terms of color differences. To allow commended by the CIE. In our simulation, we assumed an a uniform color space to be used, Luo et al. [25] built a average surround condition with three parameters (F 1, CIECAM02-based uniform color space (CAM02-UCS) by c 0.69, and Nc 1.0). The adaptation luminance LA (in making the following modifications to the CIECAM02 light- cd∕m2), which is often taken to be 20% of the luminance of the ness and colorfulness: reference white, was calculated using the illuminance of the 0 reference white in lux, Ew, J 1.7J∕ 1 0.007J ; (16)
LA nEw∕π; (7) M0 1∕0.0028 ln 1 0.0028M : (17) where n 0.2. The source tristimulus values were converted The Cartesian coordinates a0;b0 are M0 cos h ; to cone responses Ls, Ms, and Ss using the CAT02 transform M0 h matrix. Next, the cone responses were converted to adapted sin . tristimulus responses Lc, Mc, and Sc representing the corre- sponding colors under an implied equal-energy illuminant C. Estimation of the Number of Discernible Colors The estimate of the number of discernible colors in a color reference condition: solid depends on the counting method (square-packing, ellipse-packing, or convex-hull) used. The square-packing and Lc 100D∕Ls;w 1 − D Ls; (8) convex-hull methods assume a unit cube to be one discernible color in a Euclidean color space. Sphere/ellipse packing is re-
Mc 100D∕Ms;w 1 − D Ms; (9) ported to underestimate the number of discernible colors [37]. In the square-packing method, the number of unit squares packed in each locus embodying a solid is counted [18,20]. Sc 100D∕Ss;w 1 − D Ss; (10) Estimates based on this method fluctuate, however, depend- ing on the sampling sites. Figure 5 illustrates the square- where D is a parameter characterizing the degree of adapta- packing method. The solid line is a true locus; the dotted tion. D was computed as a function of the adaptation lumi- and solid squares are squares with different sampling sites. nance LA and surround F: The number of squares in the locus differs depending on the sampling sites (13 dotted squares and 19 solid squares). D F 1 − 1∕3.6 e− LA 42 ∕92 : (11) Rather, the area can be used to approximate the number of discernible colors in the locus without such ambiguities if Theoretically, D ranges from 0 for no adaptation to 1 each locus contains many color difference unit squares. The for complete adaptation. As a practical limitation, it rarely convex-hull method is another popular method of estimating goes below 0.6. After adaptation, the cone responses the area enclosed by a locus or the volume of a solid. The Masaoka et al. Vol. 30, No. 2 / February 2013 / J. Opt. Soc. Am. A 269
Fig. 5. (Color online) Diagram of square-packing method. The solid line is a true locus; the dotted and solid squares are squares with different sampling sites. method fails, however, if the locus or solid has concavities. For example, the convex-hull area of the locus shown in Fig. 5 is overestimated to be 14. As in the square-packing and convex-hull methods, we as- sumed a unit cube to be one discernible color in a Euclidean color space because this was suitable for verifying some previous estimates and finding the source of the inconsistent figures. Assuming that the volume of a large color solid ap- proximates the number of discernible colors within the solid, we calculated the volume of optimal-color solids in the CIELAB L ;a ;b , CIECAM02 [ J;aC;bC and J;aM ;bM ], and CAM02-UCS J0;a0;b0 color spaces. First, 100 optimal- color loci were obtained at lightness values from 0.5 to 99.5 at intervals of 1 for each viewing condition and color space. Each locus consisted of 3001 optimal colors with central wavelengths between 400 and 700 nm at 0.1 nm intervals. The Fig. 6. (Color online) Volume of optimal color solids in the CIELAB approximate volume of each color solid was obtained by sum- unit cube (top) and the number of CIE94 color differences (bottom) calculated with and without the von Kries-type CATs (CAT02, HPE, ming the areas of the loci. The area of each convex or concave and BFD) embedded in front of CIELAB as a function of the CCT locus specified by the N vertices of ak;bk , k 1; 2; …;N, of the light sources (dotted curves, blackbody radiator; solid curves, was easily calculated by the shoelace algorithm, which daylight illuminant). can compute the area of a simple polygon whose vertices are described by ordered pairs in the plane as N p Σ a b − a b ∕2 a a b b j k 1 k k 1 k 1 k j , where N 1 1 and N 1 1. SC 1 k1 Cab Cab ΔCab ; (20) To verify the estimate of Wen [19], we computed the num- ber of CIE94 color differences within the optimal-color solid in the CIELAB color space. Wen sliced an optimal-color solid p S 1 k C C ΔC ; at each lightness unit and divided each slice into concentric H 2 ab ab ab (21) rings, like onion rings. Each ring, one CIE94 unit chroma thick, was chopped into noncubic dice at intervals of the where SL 1, k1 0.045, and k2 0.015. We set the para- CIE94 unit hue. Then, he counted the dice to obtain the num- metric factors equal to unity: kL kC kH 1. For simpli- ΔC ΔC ∕S ΔH ΔH ∕S ber of CIE94 color differences or the number of discernible city, we denote 94 ab C and 94 ab H ; colors within the solid. Some rings were fragmentary because Eq. (18) is then expressed as of the distorted shape of the solid, and the estimate depended q on the sampling sites. ΔE ΔL 2 ΔC 2 ΔH 2: 94 94 94 (22) We took an analytical approach. The color difference ΔE94 is represented as If the two colors have a constant lightness value and hue, s ΔL 2 ΔC 2 ΔH 2 ΔE ab ab ; ΔC 94 (18) ab kLSL kCSC kH SH ΔC p : (23) 94 1 k1 Cab ΔCab Cab q By taking the limit of Eq. (23), a differential equation for the ΔH ΔE 2 − ΔL 2 − ΔC 2; (19) ab ab ab chroma is derived as follows: 270 J. Opt. Soc. Am. A / Vol. 30, No. 2 / February 2013 Masaoka et al.
Fig. 8. (Color online) Volume of optimal color solids in the CIE- CAM02 lightness–colorfulness space as a function of the CCT of the light sources with the illuminance of the reference white of 1571 lx (dotted curve, blackbody radiator; solid curve, daylight illuminant; squares, illuminant series F and illuminant E; crosses, estimate of Martínez-Verdú et al. [20]).
Z Z C C ab ab C Δhab dC ΔA ΔH C ≈ ab ab 94 94d 94 0 0 1 k2Cab 1 k1Cab k1 ln k2Cab 1 − k2 ln k1Cab 1 Δhab · . (27) k1k2 k1 − k2
Given that the nth optimal color of N points enclosing a two- dimensional area has a hue difference Δhab n and chroma Cab n , Δhab n hab n − 1 − hab n 1 ∕2. Note that Δhab n may be negative. The number of CIE94 color differ- Fig. 7. (Color online) Volume of optimal color solids in the N – – ences in the area enclosed by a locus consisting of optimal CIECAM02 lightness chroma (top) and lightness colorfulness ΣN ΔA n h 0 h N (bottom) spaces as a function of the CCT of the light sources with colors is calculated as n 1 94 , where ab ab the illuminance of the reference white of 200, 1000, 10,000, and and hab N 1 hab 1 . The total number of CIE94 color 100,000 lx (dotted curves, blackbody radiator; solid curves, daylight differences within an optimal-color solid is calculated by sum- illuminant). ming the number of CIE94 color differences within 100 loci obtained in the CIELAB color space.
dC ΔC 1 94 lim 94 : (24) ΔC →0 dCab ab ΔCab 1 k1Cab
Next, consider that the two colors have a constant chroma of Cab∕ cos Δhab∕2 , at the midpoint of which is Cab, and a con- stant lightness value. When jΔhabj ≪ 1,
ΔHab 2Cab tan Δhab∕2 ≈ CabΔhab: (25)
Then ΔH94 is approximated as
C Δh ΔH ≈ ab ab : 94 (26) 1 k2Cab
Consider an isosceles triangle whose base is ΔH94 and height is C ; when its apex is positioned at the achromatic point Fig. 9. (Color online) Volume of optimal color solids in the CAM02- ab UCS space as a function of the CCT of the light sources with the illu- a b 0 ΔA , the number of CIE94 color differences 94 minance of the reference white of 200, 1000, 10,000, and 100,000 lx within the triangle is approximately (dotted curves, blackbody radiator; solid curves, daylight illuminant). Masaoka et al. Vol. 30, No. 2 / February 2013 / J. Opt. Soc. Am. A 271
Fig. 10. (Color online) Top view of the loci of optimal color solids in the CIELAB color space computed with and without the von Kries-type CATs (CAT02, HPE, and BFD) and in the CIECAM02 lightness–chroma space J;aC;bC under blackbody radiators at 2000 and 4000 K and daylight illuminants at 6500 and 10,000 K (1000 lx) for lightness values from 5.5 to 95.5 at intervals of 5 (instead of intervals of 1 to avoid printing too many lines).
3. RESULTS CATs (CAT02, HPE, and BFD) embedded in front of CIELAB. A. CIELAB-Based Analysis The results clearly show that the number of discernible colors Figure 6 shows the volume of optimal-color solids in the and the variation with the CCT of the adaptation light source CIELAB color space (top) and the number of CIE94 color depend on the color model used. The breaks in the curves at differences (bottom) as a function of the CCT of the light 4000 K are due to the switch from the spectral power distribu- sources, which consisted of blackbody radiators (2000– tion of the blackbody radiator to that of the daylight illumi- 4000 K) and daylight illuminants (4000–10,000 K). The vo- nant. The estimated number of colors at a CCT of 6500 K lumes were calculated with and without the von Kries-type is 2,286,919 in the CIELAB unit cube, which is close to Pointer 272 J. Opt. Soc. Am. A / Vol. 30, No. 2 / February 2013 Masaoka et al.
B. CIECAM02-Based Analysis Figure 7 shows the volume of optimal-color solids in the CIECAM02 J;aC;bC and J;aM ;bM spaces as a function of the CCT for Ew values of 200, 1000, 10,000, and 100,000 lx. Morovic et al.’s[23] estimate of 3.8 million JaCbC cubes under illuminant D50 with an Ew of about 1000 lx is considerably larger than our estimate of 2,078,589, although our result is slightly larger than Morovic’s previous estimate of 1.9 million in his book [22], despite his use of the convex-hull method. The former discrepancy is due to miscalculation on their part; the latter is due to the somewhat sparse sampling of optimal colors in his computation [39]. Figure 8 shows the results of Martínez-Verdú et al. [20] and our estimate for an Ew value of 1571 lux (LA 100). These results are similar, although their highest number under illuminant E does not match ours. Figure 9 shows the volume of optimal-color solids in the 0 0 0 CAM02-UCS J ;a;b space as a function of the CCT for Ew values of 200, 1000, 10,000, and 100,000 lx. The volume increases as the CCT and illuminance of the reference white increase, which is the same trend as in Fig. 7 (bottom), although the estimate is relatively small.
4. DISCUSSION The tendencies in the estimations are determined mainly by the CAT. As shown in Fig. 6, the volume estimated using CIELAB (without the von Kries-type CATs) peaks at around 4000 K. The estimates made using CAT02 increase as the CCT increases, whereas those made using HPE and BFD decrease. The CIECAM02 estimates (Fig. 7) increase as the CCT in- creases, which is reasonable because the CAT02 chromatic adaptation transform is the main component of CIECAM02. On the whole, the estimates of standard illuminants shown in Fig. 8 also increase with the CCT, despite the dependence on the illuminant spectra. The CAM02-UCS estimates also in- crease with the CCT for the same reason as for the CIECAM02 estimates. Figure 10 shows the top view of the loci of optimal-color solids in the CIELAB color space computed with and without the von Kries-type CATs (CAT02, HPE, and BFD) and in the CIECAM02 J;aC;bC color space under blackbody radiators at 2000 K and 4000 K and daylight illuminants at 6500 K and 10,000 K (1000 lx). Figure 11 shows the corresponding hue angular volume. CAT02 and CIECAM02 exhibit essentially the same trends in the hue angular volume: the yellow component increases as the CCT increases, although the shapes of the loci differ depending on the chromatic adaptation model. The Fig. 11. (Color online) Hue angular volume of optimal color solids in matrix coefficients of CAT02 in Eq. (4) and BFD in Eq. (6) the CIELAB color space computed with and without the von Kries- are relatively close to each other, whereas the shapes of the type CATs (CAT02, HPE, and BFD) and in the CIECAM02 lightness– loci differ. These results indicate that a slight difference in the chroma space J;aC;bC under blackbody radiators at 2000 and 4000 K and daylight illuminants at 6500 and 10,000 K (1000 lx). coefficients of the CAT matrices produces different trends in volume estimation. As shown in Fig. 7, the estimates in the CIECAM02 and Attridge’s[18] estimate of 2.28 million, and the number of lightness–chroma space J;aC;bC and lightness–colorfulness CIE94 color differences is 351,791, which is close to Wen’s space J;aM ;bM increase as the CCT increases. However, the [19] estimate of 352,263. These estimates are very sensitive former is approximately independent of the adaptation lumi- to the limen chosen [38]. Considering that one JND corre- nance level, whereas the latter increases with the illuminance sponds to half of the CIE94 unit, the estimate for the number of the reference white Ew. This is reasonable from the rela- of discernible colors should be larger than the number of tionship between chroma and colorfulness described in CIE94 color differences by a factor of eight. At 6500 K, the Eq. (15). When LA ≫ 1, the fourth power of k in Eq. (14)is volume is estimated to be 2,110,746, which is again close to approximated as zero, and Eq. (13) is then reduced 1∕3 1∕3 Pointer and Attridge’s estimate. to F L ≈ 0.1 5LA 0.1 Ew∕π . Then M 0.79C for Ew Masaoka et al. Vol. 30, No. 2 / February 2013 / J. Opt. Soc. Am. A 273
Fig. 12. (Color online) Chromaticity distributions of the Munsell matte colors in the CIELAB color space computed with and without the von Kries-type CATs (CAT02, HPE, and BFD) and in the CIECAM02 lightness–chroma space J;aC ;bC under blackbody radiators at 2000 and 4000 K and daylight illuminants at 6500 and 10,000 K (1000 lx). The marker color represents the Munsell matte colors simulated under illuminant D65. values of 200 lx, M 0.91C for 1000 lx, M 1.10C for color space in terms of color differences. The significant de- 10,000 lx, and M 1.33C for 100,000 lx. The volume ratios pendence of the volume estimation on the illuminance of in lightness–colorfulness to lightness–chroma are then the reference white is, however, not taken into account. The roughly 0.63∶1 for 200 lx, 0.83∶1 for 1000 lx, 1.21∶1 for CIECAM02 lightness–colorfulness space was selected for the 10,000 lx, and 1.78∶1 for 100,000 lx. base color space of CAM02-UCS because its performance fac- It might be reasonable to use CAM02-UCS rather than tor measure (PF∕3)[40] was slightly better in the lightness– CIECAM02 for the estimation because CIECAM02 was origin- colorfulness space than in the lightness–chroma space. ally designed as a color appearance model, not a uniform However, the combination of lightness and colorfulness is 274 J. Opt. Soc. Am. A / Vol. 30, No. 2 / February 2013 Masaoka et al.
needed to determine which color space is reasonable for computing the volume of color solids. Morovic et al. [23] noted that any color appearance model fails to estimate the number of discernible colors within an optimal-color solid because the gamut is larger than the psy- chophysical data it is derived from: for CIECAM02, the LUTCHI data [24] were used. CIELAB has a limitation rooted in its development [41]: it was optimized on the basis of the Munsell system, which is defined only for the CIE 1931 stan- dard observer and illuminant C [42]. We used a dataset [43] consisting of the reflectance spectra of 1269 color chips from the matte edition of the Munsell Book of Color as a set of col- ors with a smaller gamut. Figure 12 shows the chromaticity distributions of the Munsell colors in the CIELAB color space computed with and without the von Kries-type CATs (CAT02, HPE, and BFD) and in the CIECAM02 lightness–chroma space J;a ;b Fig. 13. (Color online) Convex-hull volume of Munsell matte colors C C under blackbody radiators at 2000 and 4000 K and in the CIELAB color space calculated with and without the von Kries- daylight illuminants at 6500 and 10,000 K (1000 lx). Figure 13 type CATs (CAT02, HPE, and BFD) and in the CIECAM02 lightness– shows the convex-hull volume of the Munsell color solid in J;a ;b chroma space C C as a function of the CCT of the light sources the CIELAB unit cube calculated with and without the von (1000 lx) (dotted curves, blackbody radiator; solid curves, daylight illuminant). Kries-type CATS (CAT02, HPE, and BFD), as well as in the CIECAM02 JaCbC unit cube, as a function of the CCT of the light sources. The tendencies of increasing and decreasing unreasonable because lightness is a relative correlate and again depend on the CAT. colorfulness is an absolute correlate [33]. It seems to be nat- If it is assumed that the goal of chromatic adaptation is to ural in terms of matching the type of scale to use brightness– maximize color constancy, the adaptation model with the flat- colorfulness or lightness–chroma, although further study is test curve is probably the best. If instead it is assumed that
Fig. 14. (Color online) Chromaticity distributions of the Munsell matte colors in the CIECAM02 lightness–chroma space J;aC ;bC under black- body radiators at 2000 and 4000 K and daylight illuminants at 6500 and 10,000 K (1000 lx) with D factors of 1 (complete adaptation), 0.8, and 0.6 (practical lower limit of incomplete adaptation). The marker color represents the Munsell matte colors simulated under illuminant D65. Masaoka et al. Vol. 30, No. 2 / February 2013 / J. Opt. Soc. Am. A 275 chromatic adaptation is about maximizing the number of perceptible colors overall, then maximizing the area might be better (especially at low CCTs and low luminance levels at the beginning and end of the day), and the model with the max- imum area might be better. It would be prudent to consider them both to be partial factors. However, it is probably safest to assume that the gamut volume and chromatic adaptation are related only incidentally. Figure 14 shows the chromaticity distributions of the Munsell colors in the CIECAM02 lightness–chroma space J;aC;bC under blackbody radiators at 2000 and 4000 K and daylight illuminants at 6500 and 10,000 K (1000 lx) with D factors of 1 (complete adaptation), 0.8, and 0.6 (practical lower limit of incomplete adaptation). Figure 15 shows the volume of the Munsell colors in the CIECAM02 JaCbC unit cube as a function of the CCT of the light sources. The shapes Fig. 15. (Color online) Convex-hull volume of Munsell matte colors of the curves representing the volume calculated with CAT02 in the CIECAM02 lightness–chroma space J;aC ;bC with D factors of in Fig. 13 and CIECAM02 (D 1) in Fig. 15 resemble each 1 (complete adaptation), 0.8, and 0.6 (practical lower limit of incom- D 0 6 D 0 8 plete adaptation) as a function of the CCT of the light sources other. The volume with . and . in CIECAM02, (1000 lx) (dotted curves, blackbody radiator; solid curves, daylight however, increases unrealistically with the CCT, which illuminant). indicates a defect in the CAT. Martínez-Verdú et al. [20] suggested the possibility of an alternative color-rendering index based on the number of dis- To solve the conundrum, we must find a non-von Kries-type cernible colors within an optimal-color solid. As shown in CAT or a completely new color appearance model, but that is Fig. 8, however, their highest number under illuminant E well beyond the scope of this paper, or indeed any single did not match ours because of miscalculation on their part paper. There are no documents that clearly and correctly de- [44]. In either case, the volume estimation depends strongly scribe the applicable range of the von Kries-type chromatic on the color appearance model used, and the ranking is adaptation in terms of color temperature change for this type merely an artifact of CIECAM02. Pinto et al. [45,46] conducted of computation. In fact, von Kries had fairly low expectations subjective experiments to determine the CCT of daylight of his ideas on chromatic adaptation [35], but his work has illumination preferred by observers when appreciating art stood up rather well over the past 100 years. Although our re- paintings. They computed the chromatic diversity under search shows negative results, we believe it is a significant daylight illuminants with CCTs of 3600–25,000 K in the step in the advancement of color science. CIELAB color space and counted the number of nonempty unit cubes occupied by the corresponding color volume. They found that the distribution of observers’ preferences had a maximum at a CCT of about 5100 K and suggested that this 5. CONCLUSIONS preference has a positive correlation with the number of We estimated the number of discernible colors over a wide discernible colors. From Figs. 6 and 13, where the volume range of color temperatures and illuminance levels using sev- estimated with CIELAB peaks at around 4000 K, the match eral color models and fast, accurate methods and verified the between the preference and the number of discernible previous estimates; some estimates are almost the same as colors is, however, undeniably coincidental. Masuda and ours, and the others are miscalculations on the part of their Nascimento [47] investigated illuminant spectra maximizing authors. Our comprehensive simulation revealed that the es- the theoretical limits of the perceivable object colors. They timates depend strongly on the color model used. The varia- computed the volume of the optimal-color solid in the CIELAB tions in the volume as a function of the color temperature color space under a large number of metameric illuminants at were determined mainly by the CAT, and slight differences CCTs of 2222–20,000 K and identified a maximum volume at in the coefficients of the CAT matrices caused different trends around 5700 K. They suggested that the estimate of Martínez- in the estimation. The applicable range of the von Kries-type Verdú et al. that illuminant E (CCT 5460 K) ranked first in chromatic adaptation is unknown in terms of the color tem- gamut volume, and the positive correlation between a CCT of perature change for this type of computation. In particular, the 5100 K and observers’ preferences found by Pinto et al. were, D factor or degree of adaptation incorporated in the CAT ma- along with their findings, possible grounds for the use of the trices has an unnatural effect on the estimate. In addition, the standard illuminant D50 in the printing industry. Quintero et al. choice between the absolute correlate and the relative corre- [48] suggested that colorfulness evaluated on the basis of late significantly affects the dependence of the volume estima- the volume of optimal-color solids complements the general tion on the illuminance level. As far as the ranking of the color-rendering index based on color fidelity. However, as number of discernible colors is concerned, further research described in Section 2, CIELAB has an intrinsic limitation is required to compare the estimates at different color tem- in the applicable range of its adaptation transform, so compar- peratures and illuminance levels, even if the gamut is small. isons of the volume estimated using CIELAB at different color Thus, the number of discernible object colors remains a temperatures are not reliable. conundrum. 276 J. Opt. Soc. Am. A / Vol. 30, No. 2 / February 2013 Masaoka et al.
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